According to the principles of special relativity, information cannot be transmitted faster than the speed of light. Therefore, the concept of "instantaneous" determination or communication of the spin of an entangled particle between observers moving relative to each other is not supported by our current understanding of physics.
Let's consider the scenario you described: a train moving at near light speed with two entangled particles along its path. If one of the entangled particles is observed, and its spin is measured by a stationary observer, according to our current understanding, the measurement of the spin on one particle will not instantaneously determine the spin of the other particle as perceived by an observer on the train.
From the perspective of an observer on the train, the measurement of the spin of one particle would occur within their frame of reference, and they would observe it as a local event. They would not witness any instantaneous change or correlation with the other particle's spin.
To understand this better, we need to consider the principles of relativity and quantum mechanics. Quantum entanglement is a phenomenon where two or more particles become correlated in such a way that the state of one particle is dependent on the state of the other, regardless of the distance between them.
However, the measurement or observation of one entangled particle's spin does not determine the spin of the other particle instantaneously, regardless of the observers' relative speeds. The measurement outcomes of the entangled particles are probabilistic and independent of each other until they are compared or communicated through traditional means, which obey the speed of light as an upper limit for information transmission.
Therefore, while the observers on the train and the stationary observer may measure the spins of the entangled particles differently due to their relative velocities, the measurement outcomes will still follow the probabilistic nature of quantum mechanics, and the information will be transmitted at or below the speed of light.